Another big thing to note in examples like e.g. iteratively computing a square root for the quadratic formula or iteratively computing eigenvalues to solve a matrix: the optimization problems we’re solving are subproblems, not the original full problem. These crucially differ from most of the examples in the OP in that the system’s objective function (in your sense) does not match the objective function (in the usual intuitive sense). They’re iteratively optimizing a subproblem’s objective, not the “full” problem’s objective.
That’s potentially an issue for thinking about e.g. AI as an optimizer: if it’s using iterative optimization on subproblems, but using those results to perform some higher-level optimization in a non-iterative manner, then aligning the sobproblem-optimizers may not be synonymous with aligning the full AI. Indeed, I think a lot of reasoning works very much like this: we decompose a high-dimensional problem into coupled low-dimensional subproblems (i.e. “gears”), then apply iterative optimizers to the subproblems. That’s exactly how eigenvalue algorithms work, for instance: we decompose the full problem into a series of optimization subproblems in narrower and narrower subspaces, while the “high-level” part of the algorithm (i.e. outside the subproblems) doesn’t look like iterative optimization.
Another big thing to note in examples like e.g. iteratively computing a square root for the quadratic formula or iteratively computing eigenvalues to solve a matrix: the optimization problems we’re solving are subproblems, not the original full problem. These crucially differ from most of the examples in the OP in that the system’s objective function (in your sense) does not match the objective function (in the usual intuitive sense). They’re iteratively optimizing a subproblem’s objective, not the “full” problem’s objective.
That’s potentially an issue for thinking about e.g. AI as an optimizer: if it’s using iterative optimization on subproblems, but using those results to perform some higher-level optimization in a non-iterative manner, then aligning the sobproblem-optimizers may not be synonymous with aligning the full AI. Indeed, I think a lot of reasoning works very much like this: we decompose a high-dimensional problem into coupled low-dimensional subproblems (i.e. “gears”), then apply iterative optimizers to the subproblems. That’s exactly how eigenvalue algorithms work, for instance: we decompose the full problem into a series of optimization subproblems in narrower and narrower subspaces, while the “high-level” part of the algorithm (i.e. outside the subproblems) doesn’t look like iterative optimization.